Computer Science and Artificial Intelligence Laboratory
Technical Report
MIT-CSAIL-TR-2008-071
December 2, 2008
Resilient Knowledge-Based Mechanisms
For Truly Combinatorial Auctions (And
Implementation in Surviving Strategies)
Jing Chen and Silvio Micali
m a ss a c h u se t t s i n st i t u t e o f t e c h n o l o g y, c a m b ri d g e , m a 02139 u s a — w w w. c s a il . mi t . e d u
Resilient Knowledge-Based Mechanisms For Truly Combinatorial Auctions
(And Implementation in Surviving Strategies)
Jing Chen and Silvio Micali
CSAIL, MIT
{jingchen, silvio}@csail.mit.edu
November 17, 2008
Abstract
We put forward a new mechanism achieving a high benchmark for (both revenue and) the sum of revenue
and efficiency in truly combinatorial auctions. Notably, our mechanism guarantees its performance
• in a very adversarial collusion model;
• for any profile of strategies surviving the iterated elimination of dominated strategies; and
• by leveraging the knowledge that the players have about each other (in a non-Bayesian setting).
Our mechanism also is computationally efficient, and preserves the players’ privacy to an unusual extent.
1
Introduction
The Problems of Traditional Mechanism Design. Traditional mechanism design achieves a desired
property P “at equilibrium” and suffers from two main problems: (1) Equilibrium Selection and (2) Collusion.
The first problem is connected with the existence of multiple “reasonable” equilibria. Indeed, even if the
game is designed so that P holds at each of the possible equilibria, if some players believe that the equilibrium
which will be played out is σ while others believe that it is τ , then the profile of strategies actually selected
will not be an equilibrium at all (but rather a mix and match of σ and τ ), and P may not hold.
Equilibrium selection ceases to be a problem when P is achieved in dominant strategies, that is when P
holds at an equilibrium σ such that, for all player i, σi is the best strategy for i no matter what strategies
the other players may select. (In this case, in fact, whether or not other equilibria exist, one can confidently
predict σ to be the one actually played out by rational players.) Even so, however, P may not be guaranteed
at all in the presence of collusion. Indeed equilibria (even dominant-strategy ones) are very fragile notions:
they only imply that no single player has any incentive to deviate from his envisaged strategy, but two or
more players may have all the incentives in the world to jointly deviate from their equilibrium strategies.
The aim of this paper is to find new solutions to these fundamental problems for the case of truly
combinatorial auctions1 .
Truly Combinatorial Auctions and the Power of Collusion. Truly combinatorial auctions are very
general: there are multiple goods for sale and any player may value any subset of them for an arbitrary
amount. These general auctions are quite difficult to work with. (To keep their complexity “confined”, we
assume that no resale of the goods is allowed.) As for all auctions, the traditional goals are to maximize
revenue or economic efficiency. However, no revenue guarantee for truly combinatorial auctions was known
until this conference [MV’08]. As for economic efficiency, it is achieved in dominant strategies by the famous
VCG mechanism [V’61, C’71, G’73], but in a way that is totally vulnerable to collusion. Indeed, as shown
by Ausubel and Milgrom [AM’06], even two minimally informed collusive players, who minimally value the
goods for sale, can coordinate their bids so that the mechanism assigns to them all goods for a price of 0.
Resilient Mechanism Design and Its Limitations. In [MV’08] a mechanism is called resilient if it
guarantees its desired property without any equilibrium-selection and/or collusion problems. In their framework, collusion is unrestricted, but nothing is expected from collusive players: that is, ideally all collusive
players disappear by magic, leaving whatever mechanism to be run with just the independent players. Thus
the designer of a mechanism has no responsibility if all players are collusive, but is fully responsible so long
as a single independent player exists. We adopt the same principle. Technically, focusing on the case of a
revenue oriented auction, this principle translates into the fact that a mechanism should aim at guaranteeing
the highest possible fraction of a revenue benchmark B evaluated on just the bids of the independent players.
(Of course, the mechanism should provide such a guarantee without knowing which players are independent,
by solely relying on a properly designed incentive structure.)
Although providing a resilient mechanism for truly combinatorial auctions, [MV’08] also provides a very
draconian upperbound on the revenue that can be guaranteed in a truly combinatorial auction by any
dominant-strategy truthful (DST) mechanism. Essentially, letting MSW−? denote the benchmark consisting
of the maximum social welfare after removing the “star” player (that is the one valuing some subset of the
goods more than anyone values any subset)2 , they prove that
1
In a combinatorial auction context, there are n players and a set G of m goods. Each player i has a valuation for the goods
—a mapping from subsets of G to nonnegative reals—, denoted as T Vi . The profile (i.e., a vector indexed by the players) T V
is called the true valuation profile of the auction. An outcome consists of: (1) an allocation A, that is, a partition of G into
n + 1 subsets, A = (A0 , A1 , . . . , An ), and (2) a price profile P , that is, a profile of real numbers. A0 is referred to as the set of
unallocated goods, Ai is the set of goodsP
allocated to i, and Pi is the price of i. Relative to an outcome
(A, P ), the social welfare
P
is defined by the function sw(A, T V ) = i T Vi (Ai ), and the revenue is defined by rev(A, P ) = i Pi .
2
For any valuation (sub)profile V , letting msw(V ) = maxA sw(A, V ), and letting the “star” player, ?, be defined as ? =
1
For any truly combinatorial auction with n players and m goods, and any DST mechanism M , there exists
−? (BID)
a bid profile BID such that the (expected) revenue of M (BID) is O( MSW
log min{n,m} ).
This revenue bound is relevant to understand the power available to resilient mechanisms. Indeed, although the best way to ban equilibrium-selection problems consists of designing a DST mechanism, if one
wants to guarantee more revenue than a logarithmic fraction of MSW−? , then their bound implies that one
has only two alternatives:
A1. Assuming that more knowledge is available (e.g., that the seller has some Bayesian information about
the players’ true valuations) or
A2. Adopting a solution concept weaker than dominant strategies.
Since maximizing revenue is one of our goals, to bypass the revenue upperbound of [MV’08], it is necessary
for us to take at least one of these alternatives. Actually, in this paper we take both, but without violating the
basic principle of mechanism design (i.e., “all knowledge resides with the players”) and without introducing
any equilibrium-selection problem.
1.1
Our Contributions
New Goals. The goal of this paper is to guarantee both high revenue and high total performance in any
truly combinatorial auction. By “total performance” we mean the sum of social welfare and revenue. Since
maximizing revenue is well understood, let us clarify how our second goal differs from (and may often be
preferable to) the classical goal of maximizing social welfare.
The traditional motivation behind the maximization of social welfare is that of a benevolent government,
solely interested in the happiness of its citizens, rather than in revenue. As already mentioned, in absence
of collusion, the VCG mechanism perfectly achieves this classical goal. In doing so, the VCG imposes prices
to the players, but such prices are almost an “after thought,” or a “necessary evil:” they are just means to
maximize social welfare. But what is wrong with revenue? Even a benevolent government could transform
it into roads, hospitals and other infrastructure from which everyone benefits. Taking this point of view, in
addition to seeking resilient mechanisms for revenue, we also seek resilient mechanisms for the sum of social
welfare and revenue.
Note that, in light of the cited counterexample of Ausubel and Milgrom, the VCG mechanism is not
adequate for total performance, and for two reasons. First, even in absence of collusion, the VCG is not
optimal for this less traditional goal: in a sense, it achieves it only “within a factor of 2,” because it can
maximize social welfare returning 0 revenue. Second, in the presence of collusion, both the VCG’s returned
social welfare and revenue can be (essentially) 0.
Note too that, although in a rational setting “revenue lowerbounds social welfare,” a resilient mechanism
aiming at maximizing revenue may not maximize the sum of revenue and social welfare. This is so because,
in order to guarantee revenue in the presence of collusive players, the mechanism may have to give up some
efficiency. Consequently, the social welfare of a resilient revenue-oriented mechanism M may be exactly equal
to M ’s revenue, so that the total social-welfare-plus-revenue is just twice a modest revenue. However, by
directly aiming at maximizing total performance, a resilient mechanism may actually do much better.
A New, Knowledge-Based, Benchmark. Designing mechanisms achieving our goals might be easier
with some information about the players’ true valuations, but acquiring this information may be too hard.3
We thus assume that the designer has no knowledge whatsoever about the players, but that the players have
some knowledge about each other. In our setting each player i not only has internal knowledge, that is
knowledge of his own true valuation T Vi , but also some external knowledge, that is some information about
arg maxi maxS⊆G Vi (S), then MSW−? (V ) = msw(V−? ).
3
In particular, for auctions of a single good, Cremer and McLean [CM’88] have fully captured the information structure
needed to generate the maximum possible revenue, but concluded that acquiring this information would be too difficult for their
result to be of practical use.
2
T V−i , the other players’ true valuations. (This is without any loss of generality, since the external knowledge
of i may be “empty.”) While a player i’s external knowledge could be very general, and is indeed denoted
by GKi , the relevant knowledge, denoted by RKi , is “how well he can sell the goods to the other players”
via take-it-or-leave-it offers. (That is, RKi is an outcome for an auction whose only players are those in −i,
where everyone only pays if he receives some goods, and pays no more than his true value for the received
goods.) That is, we benchmark the performance of a totally ignorant seller with the revenue obtainable by
the best informed independent player. Again, our benchmark focuses on independent players because we too
do not count on collusive ones to achieve our goals, and consider ourselves fortunate if they spontaneously
leave the auction.
A New and General Collusive Model. We envisage a very adversarial collusion model. In particular,
we allow any number of collusive players, we let them form any number of —disjoint— collusive sets, we
do not restrict the cardinality of collusive sets, and we do not restrict the way in which the members of a
collusive set coordinate their actions. (If they so want, the members of a collusive set may enter binding
agreements on how to act.)
We insist, however, that all players be rational. As usual, an independent player, one not belonging to
any collusive set, is individually rational, that is he acts so as to maximize his individual utility function
ui , mapping each possible outcome (A, P ) to the real value T Vi (Ai ) − Pi . A collusive set C is collectively
rational, that is its members coordinate their actions so as to maximize their own collective utility function
uC , mapping any outcome (A, P ) to a real number.
To maximize meaningfulness, we want the relationship between uC and the individual utility functions of
C’s members to be as general as possible, provided that we do not “transform collusive players into irrational
ones.” (Indeed, what is the difference between saying that (1) C is a set of crazy players and (2) C is a set of
rational players acting to maximize a crazy collective utility function uC ?4 ) Accordingly, we demand that uC
be minimally monotone. Let us explain. Consider two outcomes that are absolutely identical, as far as C’s
members are concerned, except for member i who receives no goods and pay nothing in the first outcome,
but receives a subset of goods Ai for a price Pi in the second one. Then, minimal monotonicity requires that
C prefer the second outcome if Pi < T Vi (Ai ).5
Minimal monotonicity is of course a restriction on C’s collective utility function, but: (a) it is the only
restriction to our otherwise general collusion model; and (b) it is a very reasonable restriction.6
A New Solution Concept. Although not DST, the mechanism designed in this paper is immune to any
equilibrium-selection problem. The reason is very simple: we rely on an equilibrium-less solution concept. In
essence, our mechanism guarantees that, as long as each player selects a strategy surviving iterated elimination
of weakly dominated strategies, our goal is achieved. We call such a solution concept implementation in
surviving strategies. After weakly dominated strategies are removed iteratively, each player is left with a
plurality of surviving strategies, and ultimately he chooses one of them to play. Thus, with an implementation
in surviving strategies, it is quite possible that the profile of strategies actually played is not an equilibrium
at all. Yet, the desired property is guaranteed just the same: any profile of “not-dumb” strategies will do.
4
Indeed, irrational players may be modeled as taking arbitrary (i.e., universally quantified) actions, and for any tuple of
actions actually taken by the members of C, one might be able to find an ad hoc collective utility function uC so as to rationalize
their actions as maximizing that uC .
5
For example, a minimally monotone uC may consist of the sum of the individual utilities of C’s members. As for a more
eccentric example, uC may be the sum of: the individual utility of C’s first member, half of the utility of C’s second member, a
third of the individual utility of C’s third member, and so on.
6
In a sense, since each of them receives exactly the same goods for exactly the same price, the other members of C —if
consulted when choosing uC — should have no reason to object against i’s receiving goods that he values more than he pays. If
side-payments were possible, then they could request additional compensation from a happier i. And if side-payments were not
possible, once they have the same goods and pay the same price, then under traditional economic models “making i unhappier
would not make them happier.” In any case, mechanism design presupposes the players’ rationality, and as we said some formal
restriction is needed to prevent collusion from becoming de facto indistinguishable from irrationality.
3
Therefore, differently from a Nash equilibrium, our solution concept does not rely at all on the players’
beliefs, but only on their rationality. As for another point, in our implementation it is hard to predict
precisely which profile of strategies will be ultimately played. But while “strategy predictability” has always
been the cornerstone of traditional mechanism design, it has always been a mean to an end, not the end
itself. Ultimately, in an auction,
we do not care about predicting strategies, but we care a lot about predicting revenue and efficiency.
Our solution concept generalizes the one of implementation in undominated strategies, even in absence
of collusion, and applies to the presence of collusion as well. (See Section 2 for details.)
Our Mechanism. In sum, we exhibit a very resilient mechanism M for truly combinatorial auctions. As
long as the players select strategies surving iterated elimination of dominated strategies, M guarantees, under
a very general collusion model and without any information about the players, a total performance within a
factor of 2 of the revenue obtainable by the best informed independent player.
2
Prior Work
A large body of work has been devoted to protect auctions against collusion. Notably, [JV’01, MS’01, FPS’00]
proposed group strategy-proof mechanisms, which are robust against collusive players unable to make sidepayments to each other. By contrast, we do not envisage such restrictions in our paper. (In particular, we
allow collusive players to make payments to one another and/or enter secret and binding contracts.)
The paper of [GH’05] proposes the notion of a c-truthful mechanism, for which no collusive set with at
most c members can collectively gain more than what they could get by acting individually. However, the only
mechanisms satisfying this notion are those that, independent of any bids, offer any subset of goods S to any
player i for a fixed price pS,i . Such mechanisms, therefore, are far from maximizing revenue if one does not have
a proper Bayesian information about the players’ true valuations for the goods. In the same paper the authors
also propose mechanisms that are free to choose more general outcomes (so as to approximate maximum
revenue), but satisfy a weaker notion of collusion resilience and apply to restricted auctions: namely, single
good in unlimited supply. By contrast, we do not put any restriction on the cardinality of the collusive sets,
nor restrict the types of combinatorial auctions. Other notable resilient mechanisms for a variety of restricted
auctions are due to [LOS’02, BLP’06, BBM’07, GHKKKM’05, LS’05, FGHK’02, GHKSW’06, BBHM’05].
Once more, however, we do not rely on any auction restriction.
We also note that all the above mentioned papers aim at guaranteeing high efficiency and/or revenue,
but not —as we do— total performance. Further, the benchmarks of all these prior works are expressed in
terms of the players’ valuations. As far as we know we are the first to work with a benchmark based on the
players’ knowledge.
Finally, all prior works mentioned above adopt dominant-strategies as a solution concept. A solution
concept relevant to ours is the classical one of implementation in undominated strategies. In essence, in
our language, a mechanism M achieves a property P in undominated strategies if P holds for any outcome
obtained by running M on a profile of undominated strategies. (See Jackson [J’92] for a formal version.)
This notion, however, was never exemplified in any setting of incomplete information, let alone in auctions.
Babaioff, Lavi and Pavlov [BLP’06] both proposed a feasible variant of this notion (in essence, each player
can compute his undominated strategies efficiently) and provided the first (and efficient too) mechanism
satisfying it for a restricted type of combinatorial auctions. Namely, their mechanisms applies to auctions
in which each player i has only two possible values for any subset of the goods: either 0 or a fixed value vi .
In sum, the solution concept of [BLP’06] requires less rationality than ours, but their mechanism does not
address collusion at all, and does not apply to truly combinatorial auctions.
4
3
Our Knowledge Benchmark
Recall that we denote by GKi the general external knowledge of a player i, and by RKi i’s relevant external
knowledge, properly deduced from (and thus compatible with) GKi . Let us consider a few examples.
1. GKi consists of a subset of V−i (the set of all possible valuation sub-profiles for the players in −i) such
that T V−i ∈ GKi . Here GKi represents the set of possible candidates, in i’s opinion, for the other
players’ true valuations. Such GKi is genuine in the sense that one of its candidates is the “right one.”7
In this example, RKi is deduced from GKi in two conceptual steps. First, one computes all outcomes
(A, P ) feasible for GKi , that is the outcomes with integer prices such that, for all players j ∈ −i and
all valuations subprofiles V ∈ GKi , Pj < Vj (Aj ). Then, the relevant external knowledge RKi consistent
with GKi is the outcome with maximum revenue among the outcomes feasible for GKi . (Thus, if
GKi = V−i , then RKi is the null outcome.)
2. GKi consists of a probabilistic distribution over V−i that assigns positive probability to the actual T V−i .
In this case, RKi is the outcome with the maximum revenue among all those outcomes feasible for the
support of GKi .
3. GKi consists of a “partial” probability distribution over V−i . For instance, starting with a distribution
D assigning positive probability to the actual subprofile T V−i , GKi is derived from D as follows: when
the probability pV of each subprofile V ∈ V−i is positive, then pV is replaced with a subinterval IV of
[0, 1] that includes pV . (IV = [0, 1] is interpreted as i knowing “nothing” about profile V .) In this case,
the outcomes consistent with GKi are those feasible for the set of subprofiles V whose subinterval does
not coincide with [0, 0]. And among such outcomes, RKi is the one whose revenue is maximum.
In sum, alongside with the true-valuation profile T V , we consider the profiles GK and RK to be integral
components of the original context of any combinatorial auction.
An Important Clarification. It is crucial to clarify that, although our benchmark is solely based on
the relevant external knowledge of the players, our results do not assume that the players’ sole knowledge
is the one relevant to our benchmark (i.e., that GKi = RKi for all i), nor that each GKi has a specific
form. Such assumptions may be very convenient for designing mechanisms, but very unrealistic for running
these mechanisms. Indeed, we expect the players to have all kinds of external knowledge in addition to the
one relevant to us, and to rationally act relying on all the knowledge available to them once a mechanism is
chosen. Accordingly, to enhance the meaningfulness of our results, we do not restrict the players’ external
knowledge at all. That is,
Our mechanism achieves our RKi -based benchmark for all possible GKi s consistent with the RKi s.
(We are fully aware, of course, that better performance could be guaranteed by assuming some suitable
restriction for the players’ external knowledge.) Let us now be more precise.
Definition 1. (External, Canonical and Feasible Outcomes.) Let (A, P ) be an auction outcome. We
say that (A, P ) is external for a player i, if Ai = ∅ and Pi = 0. We further say that such (A, P ) is canonical
external for i if, ∀j 6= i, Pj = 0 whenever Aj = ∅, and a positive integer otherwise. We further say that such
(A, P ) is feasible external for i, relative to a valuation profile V , if ∀j 6= i, Pj < Vj (Aj ) whenever Aj 6= ∅.
Notice that a feasible external outcome (A, P ) for i, relative to the true-valuation profile T V , corresponds
to a simple and guaranteed way of selling the goods to the players in −i. Namely, offer the subset of goods
Aj to player j for price Pj : if j accepts the offer, he will receive the goods in Aj and pay Pj ; else j pays
nothing and receives no goods. Such a way of selling the goods is guaranteed to succeed if the players are
rational. Indeed, since each non-empty subset of goods is offered at an “attractive” price, each player offered
some goods should rationally accept the offer.
7
Notice that GKi = V−i expresses the fact that i knows “nothing” about T V−i . Also notice that a proper choice of GKi can
precisely express pieces of i’s external knowledge such as “player h’s valuation for subset S is larger than player j’s valuation for
subset T .”
5
Definition 2. (Original Context) The original context of a combinatorial auction is a triple of profiles,
(T V, GK, RK), where for each player i (1) T Vi is i’s true valuation; (2) GKi is the information known to i
about T V−i ; and (3) RKi is the outcome with maximum revenue among all feasible external outcomes for i,
relative to all valuation profiles V consistent with GKi .
We refer to GKi as i’s general external knowledge, and to RKi as i’s relevant external knowledge.
Notice that our relevant external knowledge is non-Bayesian. Indeed, although the general external
knowledge of a player i may naturally arise in a Bayesian setting, RKi always is a way for i to sell the goods
to the other players that succeeds with probability 1 when the players are rational.
Definition 3. (The MEW Benchmark) We let MEW, the maximum external welfare, to be the function
so defined: for any relevant-external-knowledge subprofile RKS ,
MEW(RKS ) = max rev(RKi ).
i∈S
Letting I denote the set of all independent players, we define our benchmark to be MEW(RKI ).
4
Our Collusion Model
Definition 4. Given an original context (T V, GK, RK), a minimally monotone collusive set consists of a
subset C of two or more players and a function uC from outcomes to real numbers such that, ∀i ∈ C and ∀
outcomes (A, P ) and (A0 , P 0 ) for which (1) (Aj , Pj ) = (A0j , Pj0 ) ∀j ∈ C \ {i} and (2) A0i = ∅ and Pi0 = 0:
uC (A, P ) ≥ uC (A0 , P 0 ) if and only if T Vi (Ai ) − Pi ≥ 0.
Notice that the original context specifies the knowledge of all players, including the members of C. Yet, we
do not specify how this individual knowledge is combined (if at all) to form a possible “collective knowledge”
of C. This is so for two reasons. First, such a knowledge may be hard to predict, as in principle there is no
way to guarantee that C’s members truthfully reveal their private knowledge to each other. (For instance,
if C arose from an initial negotiation, then a member i of C might have had incentives to lie about his
knowledge in order to enter C and/or influence in his favor the choice of uC .8 ) Second, we do not need any
assumption on such knowledge to achieve our results. The minimal monotonicity of uC is all we require.
Let us now formally discuss the possibility of having a plurality of collusive sets. Here, our only restriction is the disjointness of the collusive sets. (Else, discussing collective rationality would become more
problematic.) For uniformity of presentation, we specify the collusive as well as the independent players via
a partition C of the set of all players: namely, a player set C in C is collusive if it has cardinality greater
than 1, and a player i is independent if his collusive set has cardinality 1, that is if {i} ∈ C. (This way each
player i, collusive or not, belongs to a single set of C, denoted by Ci .)
Definition 5. (Minimally Monotone Collusive Contexts and Auctions) A minimally monotone
collusive context C is a tuple (T V, GK, RK, C, I, U) where
• (T V, GK, RK) is the original context of the auction.
• C is a partition of the players.
• I is the set of all players i such that {i} ∈ C (explicitly specified for convenience only).
• U is a vector of functions, indexed by the subsets in C, such that (1) UC is a minimally monotone
collective utility function if C’s cardinality is > 1, and (2) U{i} (A, P ) = T Vi (Ai ) − Pi if i ∈ I.
A minimally monotone collusive auction is a pair (C , M ), where C is a minimally monotone collusive auction
context and M is an auction mechanism.
8
Perhaps better results may be obtained by restricting the collective knowledge (or the process of coalition formation) but
these possibilities are not investigated in this paper.
6
We refer to a player in I as independent, to a player not in I as collusive, to any C ∈ C of cardinality > 1
as a collusive set and to its corresponding UC as C’s collective utility function, and U as the utility vector
of C. We use the term agent to denote either an independent player or a collusive set. For any player i, we
denote by Ci the set in C to which i belongs.
If C is a minimally monotone collusive context whose components have not been explicitly specified, then
by default we assume that C = (T V C , GK C , RK C , CC , I C , UC ).
5
Our Solution Concept
Implementation in Σ1 /Σ2I Strategies. In practice, there seem to be different levels of rationality. That
is, many players are capable of completing the first few iterations of elimination of dominated strategies, but
fail to go “all the way.” Accordingly, one should prefer mechanisms that guarantee their desired property for
any vector of strategies surviving just the first few iterations. Our mechanism achieves our benchmark for
any vector of strategies surviving the following two-round elimination process. First, each agent (i.e., each
independent player and each collusive set) removes all his weakly dominated strategies. Then, each independent player eliminates all strategies which become weakly dominated after the first round of elimination is
completed. Since the set of strategies surviving the first iteration is often referred to as Σ1 , and the set of
those surviving the first two iterations is commonly referred to as Σ2 , we call this refinement of our solution
concept implementation in Σ1 /Σ2I Strategies. For simplicity, we formalize just this latter refinement of our
solution concept, and only for our auction setting.
Difficulties with Collusion. Before formalizing our solution concept, it is worth noticing that such a
solution concept is of independent interest, and we expect it to play a larger role in perfect-information and
non-collusive settings. In such settings our notion is in fact significantly easier, because it is easy to determine
which strategies are dominated. In the present setting, instead, whether a strategy is dominated depends
on such additional factors as the collusive sets actually present and their collective utility functions, factors
about which no information is publicly available.
The presence of collusion significantly complicates both our notion and the analysis of our mechanism,
and significantly “increases” the number of surviving strategies. Indeed, since we are also dealing with players
who secretly collude and optimize secret collective utility functions, it is hard for a player or a collusive set
to “dismiss more than just a handful of strategies at each iteration.”
Formalization. Our mechanisms are of a very simple form: at each decision node all players act simultaneously, and their actions become public as soon as they are chosen. Also, our mechanisms are probabilistic,
and their coin tosses too become of public domain as soon as they are made. Finally, since in this paper we are
dealing with secret collusion, our mechanisms specify only the strategies of individual players. Accordingly,
denoting the set of all deterministic strategies of a player i by Σ0i , the set of all deterministic strategy profiles
by Σ0 , the set of all deterministic collective strategies of a collusive set C by Σ0CQ, the set of all deterministic
strategy vectors of a collusive context C by Σ0C , and the Cartesian product by , we have
Q
Q
Σ0 = i Σ0i , Σ0C = i∈C Σ0i , and Σ0C = Σ0 .9
To formalize implemention in Σ1 /Σ2I strategies, we start by adapting the standard definition of dominated
and undominated strategies to collusive auctions (where an agent is an independent player or a collusive set).
Definition 6. (Dominated, Undominated, and Σ1 Strategies in Collusive Auctions.) Let A be an
agent in a collusive auction (C , M ). We say that a deterministic strategy σA of A is dominated over a set
0 ∈ Σ0 such that
of strategy vectors Σ0 if σA ∈ Σ0A and there exists σA
A
0 tτ
1. ∀τ−A ∈ Σ0−A , E[uA (M (σA t τ−A ))] ≤ E[uA (M (σA
−A ))].
9
Indeed, for all C we have Σ0C =
Q
C∈CC
Σ0C =
Q
C∈CC
Q
i∈C
7
Σ0i = Σ0 .
0 tτ
2. ∃τ−A ∈ Σ0−A such that E[uA (M (σA t τ−A ))] < E[uA (M (σA
−A ))].
Else, we say that σA is undominated over Σ0 .
We denote by Σ1A,C the set of deterministic strategies of A undominated over Σ0C (= Σ0 ), and by Σ1C the set
Q
of strategy vectors surviving the first round of elimination of dominated strategies, that is, Σ1C = C∈CC Σ1C,C .
Definition 7. (Compatibility.) We say that a collusive context C is compatible with
• an independent player i if (1) i ∈ I C , (2) T Vi = T ViC , (3) GKi = GKiC , and (4) RKi = RKiC
• a collusive set C if (a) C ∈ CC and (b) UCC is C’s collective utility function.
Remarks. Fixing the mechanism M ,
• Σ1A,C is the same for any C compatible with A (and is computable by A). In fact, the set Σ0 is fully
determined from M alone, and which strategies of A are undominated over Σ0 solely depends on A’s
utility function (rather than, say, on the partition of the other players into collusive sets, and
Q their utility
1
1
1
functions). Accordingly, we shall more simply write ΣA instead of ΣA,C , and thus ΣC = C∈CC Σ1C .
• Σ1C is crucially dependent on C . In fact, although each C can compute Σ1C “independent of the overall
collusive context”, if C is a collusive set in one context C , it may not be a collusive set in another
context C 0 . (In C , for example, C may in particular consist of 10 players and Σ1C of just one collective
strategy. In C 0 , however, all the players in C may be independent, and for each i ∈ C, Σ1i may consist
of two individual strategies. Therefore, the players in C contribute a single strategy subvector to Σ1C ,
but 1024 strategy subvectors to Σ1C 0 .)
Definition 8. (Σ2I Strategies and Σ1 /Σ2I Plays for Minimally Monotone Contexts.) Let (C , M ) be
a minimally monotone collusive auction.
If i is an independent player in (C , M ), then we denote by Σ2i,C the set of all strategies σi ∈ Σ1i undominated over Σ1C ; and by Σ2i the union of Σ2i,C 0 for all C 0 compatible with i.
We say that a strategy vector σ is a Σ1 /Σ2I play of (C , M ) if
Y
Y
σ∈
Σ2i ×
Σ1C .
i∈I C
C∈CC ,|C|>1
Definition 9. (Implementation in Σ1 /Σ2I Strategies) Let P be a property over auction outcomes, and
M an auction mechanism. We say that M implements P in Σ1 /Σ2I strategies if, for all minimally monotone
collusive contexts C , and all Σ1 /Σ2I plays σ of the auction (C , M ), P holds for M (σ).
Remarks.
• Although σ is a vector of deterministic strategies, M may be probabilistic. In this case, M (σ) is a
distribution over outcomes, and P a property of outcome distributions.
• Precisely computing Σ2i,C requires precise knowledge of Σ1C . Accordingly, if i is unaware of the actual
collusive context C , he may be unable to compute Σ2i,C (but he could compute the much larger set Σ2i by
going through all possible collusive contexts compatible with him). In particular, i may not even think of
collusion, and thus behave as if all players were independent. However, no matter what i believes about
collusion, the strategies of i that —in i’s mind!— survive the second round of elimination of dominated
strategies will be a subset of Σ2i . Thus if a property P is implemented in Σ1 /Σ2I strategies, then it holds
no matter what strategies each independent player i believes to be his Σ2i strategies.
8
6
Our Mechanism
Although requiring some modifications, the basic idea behind our mechanism is very simple: each player i,
simultaneously with the others, announces an canonical external outcome Ωi for i, and then we run a “secondprice auction.” Let us explain. Let R0 be the second highest of all revenues of the announced outcomes, and
let ? be the “star player”, that is the one who has announced the outcome with the highest revenue. Then,
we would like to (1) sell the goods according to Ω? , so as to generate revenue R? =rev(Ω? ), and (2) divide
this revenue as follows: the seller gets R0 and the star player gets R? − R0 . To implement the first step, let α?
and π ? respectively be the allocation and the price profile of Ω? . Then we ask each player i ∈ −?, receiving
some goods in α? , whether he is willing to buy the subset of goods αi? for price πi? . If i agrees, the subsale is
final. Else, the star player pays a fine equal to πi? and the goods in αi? remain unallocated.
With this way of proceeding, it can be shown that our benchmark is achieved if each independent player
i does not “underbid”, that is, if he announces either his relevant-external-knowledge outcome RKi or an
outcome whose revenue is higher than that of RKi . But: can we be sure that an independent player i does
not underbid? The problem is that, depending on his general knowledge GKi , underbidding may be a quite
rational option. For instance GKi can be compatible with RKi and yet assure i that (1) he is the best
informed player, that is, the revenue of RKi is higher than that of the relevant-external-knowledge outcome
of any other player; (2) the next best informed player is j; (3) the revenue of RKj is very close to that of
RKi ; and yet (4) player j is badly informed about him: that is RKj allocates to i a subset of goods Si that
i highly values for a ridiculously low price Pi . In this case, i would be better off if (a) j announced RKj and
(b) j were the star player. In sum, underbidding may be far from being a dominated strategy for i.
Our mechanism thus modifies the above basic procedure as follows. Together with RKi , each player i
also announces his favorite subset of the goods. And if i is declared the star player, then a coin toss of
the mechanism determines whether the above basic procedure takes place or i receives, for free, his favorite
subset. A precise (if tedious) analysis proves that this modification is an incentive sufficient to ensure that
underbidding will not survive the iterated elimination of dominated strategies for any independent player
i. The same cannot be said about collusive players, but then we do not rely on them for efficiency nor for
revenue. However, we must ensure that they do not hurt the achievement of our benchmark, something that
unfortunately adds significant complexity to our analysis.
(In M’s description below, real actions occur in “numbered steps” and public updates in “bulleted steps.”)
Mechanism M
• Set Ai = ∅ and Pi = 0 for each player i.
1. Each player i simultaneously and publicly announces (1) a canonical external outcome for i, Ωi =
(αi , π i ), and (2) a subset Si of the goods.
• Set: Ri = rev(Ωi ) for each player i, ? = arg maxi Ri , and R0 = maxi6=? Ri .
(We shall refer to player ? as the “star player”, and to R0 as the “second highest revenue”.)
2. Publicly flip a fair coin.
• (If Heads:) reset A? = S? and HALT.
3. (If Tails:) Each player i such that αi? 6= ∅ simultaneously and publicly announces YES or NO.
• Reset: (1) P? = P? + R0 ; (2) for each player i who announces NO, P? = P? + πi? ; and (3) for each player
i who announced YES, Ai = αi? , Pi = πi? , and P? = P? − πi? .
Let us now state our theorem, proved in the appendix.
Theorem 1. For all minimally monotone collusive contexts C and all Σ1 /Σ2I plays σ of (C , M),
E[rev(M(σ))] + E[sw(M(σ), T V )] ≥
MEW(RKI )
2
9
(and E[rev(M(σ))] ≥
R0
).
2
7
Privacy and Computational Efficiency
By a proper interpretation of the revelation principle, our mechanism possesses a normal-form counterpart.
(In it, however, a collusive set C should be careful in choosing its collective strategy when the star player
is not in C; or it would essentially reveal itself as collusive.) But the revelation principle does not preserve
privacy, and privacy issues may alter the way games are played in practice, threatening the meaningfulness
of fundamental solution concepts such as dominant-strategy truthfulness. A general result of Lepinski,
Izmalkov, and Micali [ILM’08] guarantees that every normal-form mechanism has an extensive-form version
that perfectly implements it, in particular without any privacy loss. However, their construction —although
feasible— is practically inconvenient. By contrast, let us point out that mechanism M, although not designed
to achieve perfect privacy, (1) does guarantee a lot of privacy and (2) is computationally very efficient.
Putting it playfully, M is “tax-free”. Consider a second-price auction for a single good in which the winner
bids $10M and wins the item for $1M, and assume an overreaching and tyrannical tax code. Then, the IRS
may demand to collect taxes on $9M, reasoning that the winner himself, being rational in a dominant-strategy
truthful mechanism, freely admitted that he is receiving a $10M value. Such a demand cannot arise in an
Ascending English Auction (AEA). In the AEA, the players who drop out reveal their true valuations, but
are not “taxable” because they have no utility. As for the winner, he could always declare that his true value
for the object was (assuming the same valuations above) exactly $1M (plus $1 if he really feels to look more
“legitimate”). In an ascending English auction, therefore, there is nothing to “tax”.
Leaving taxes aside, there is a legitimate privacy issue here. The winner of an auction may not want to
inform its competitors of the utility he has received, and thus of his real financial strength. More generally, no
player wants to declare his own true valuation. Yet, in a combinatorial auction —even if the mechanism used
is not DST— the players themselves may provide some information or evidence about their own valuations.
By contrast, in mechanism M, no player reveals more information about himself than that “deducible from
the final outcome itself.” If M’s coin ends up Heads, then the star player receives for free his favorite subset
S, but he never said anything himself about his own valuation for S: whatever the other players say about
him is just “hearsay.” If M’s coin ends up Tails, then every player who answers YES receives goods that he
may always claim to value for exactly what he was offered to pay and indeed paid.
(Note that further privacy could be gained if M first asked each player i to announce just the revenue of
Ωi in Stage 1, and then asked only the star player to reveal both Ω? and S? . However, this alternative way of
proceeding would enable the star player to announce Ω? depending on the revenues announced by the other
players, and thus an independent player i may have incentives to underbid.)
Finally note that, unlike the VCG, M is computationally trivial. At most, it is the players who have to
work hard to deduce RKi from GKi . But this is a different matter. When an exponential-time mechanism is
(necessarily!) approximated by a computationally efficient one, crucial distortions of incentives may ensue.
But this is not the case here. If a player i imperfectly, because computationally bounded, deduces RKi from
GKi , our M still achieves our total-performance benchmark —not defined on the perfect relevant knowledge
of the players, but on the relevant knowledge actually computable by the players.
8
Extensions and Conclusions
With additional work, our mechanism can be extended so as to achieve a more demanding benchmark: the
maximum social welfare known to the independent players. (I.e., the relevant knowledge of a player now
includes the possibility of “giving himself some goods.”) Achieving this other benchmark however requires
mechanisms that also solicit separate and special bids from collusive sets. Such mechanisms were pioneered
by Micali and Valiant in a yet unpublished manuscript that has much influenced the present paper [MV’07].
In closing, we believe knowledge-based bechmarks and equilibrium-free solution concepts to be very
powerful conceptual tools that will enable us to provide meaningful and attractive solutions to a host of
other problems in mechanism design. (In particular, similar techniques can be applied to provision of a
public good.)
10
References
[AM’06] L.M. Ausubel and P. Milgrom. The Lovely but Lonely Vickrey Auction. Combinatorial Auctions,
MIT Press, pages 17-40, 2006.
[BBHM’05] M.F. Balcan, A. Blum, H. Hartline, and Y. Mansour. Mechanism Design via Machine Learning.
Foundations of Computer Science, pages 605-614, 2005.
[BBM’07] M.F. Balcan, A. Blum, and Y. Mansour. Single Price Mechanisms for Revenue Maximization in
Unlimited Supply Combinatorial Auctions. CMU-CS-07-111, 2007.
[BLP’06] M. Babaioff, R. Lavi and E. Pavlov. Single-Value Combinatorial Auctions and Implementation in
Undominated Strategies. (Full version of ) Symposium on Discrete Algorithms, pages 1054-1063, 2006.
[C’71] E.H. Clarke. Multipart Pricing of Public Goods. Public Choice, Vol.11, No.1, pages 17-33, Sep., 1971.
[CM’88] J. Cremer and R.P. McLean. Full Extraction of the Surplus in Bayesian and Dominant Strategy
Auctions. Econometrica, Vol.56, No.6, pages 1247-1257, Nov., 1988.
[FGHK’02] A. Fiat, A. Goldberg, J. Hartline, and A. Karlin. Competitive Generalized Auctions. Symposium
on Theory of Computing, pages 72-81, 2002.
[FPS’00] J. Feigenbaum, C. Papadimitriou, and S. Shenker. Sharing the Cost of Multicast Transmissions.
Symposium on Theory of Computing, pages 218-226, 2000.
[G’73] T. Groves. Incentives in Teams. Econometrica, Vol. 41, No. 4, pages 617-631, Jul., 1973.
[GH’05] A. Goldberg and J. Hartline. Collusion-Resistant Mechanisms for Single-Parameter Agents. Symposium on Discrete Algorithms, pages 620-629, 2005.
[GHKKKM’05] V. Guruswami, J.D. Hartline, A.R. Karlin, D. Kempe, C. Kenyon, and F. McSherry. On
Profit-Maximizing Envy-free Pricing. Symposium on Discrete Algorithms, pages 1164-1173, 2005.
[GHKSW’06] A. Goldberg, J. Hartline, A. Karlin, M. Saks, and A. Wright. Competitive Auctions. Games
and Economic Behavior, Vol. 55, Issue 2, pages 242-269, May, 2006.
[ILM’08] S. Izmalkov, M. Lepinski, and S. Micali. Perfect Implementation of Normal-Form Mechanisms.
MIT-CSAIL-TR-2008-028, May 2008. (A preliminary version appeared in FOCS’05, pages 585-595, with
title “Rational Secure Computation and Ideal Mechanism Design”.)
[J’92] M. Jackson. Implementation in Undominated Strategies: a Look at Bounded Mechanisms. Review of
Economic Studies, Vol. 59, No. 201, pages 757-775, Oct., 1992.
[JV’01] K. Jain and V. Vazirani. Applications of Approximation Algorithms to Cooperative Games. Symposium on Theory of Computing, pages 364-372, 2001.
[LOS’02] D. Lehmann, L. O’Callaghan, and Y. Shoham. Truth Revelation in Approximately Efficient Combinatorial Auctions. Journal of the ACM, Vol. 49, Issue 5, pages 577-602, Sep., 2002.
[LS’05] A. Likhodedov and T. Sandholm. Approximating Revenue-Maximizing Combinatorial Auctions. National Conference on Artificial Intelligence (AAAI), pages 267-274, 2005.
[MS’01] H. Moulin and S. Shenker. Strategyproof Sharing of Submodular Costs: Budget Balance Versus
Efficiency. Economic Theory, Vol.18, No.3, pages 511-533, 2001.
[MV’08] S. Micali and P. Valiant. Resilient Mechanisms for Truly Combinatorial Auctions, submitted to
Symposium on Theory of Computing, 2009.
11
[MV’07] S. Micali and P. Valiant, Leveraging Collusion in Combinatorial Auctions, Unpublished Manuscript,
2007
[V’61] W. Vickrey. Counterspeculation, Auctions, and Competitive Sealed Tenders. The Journal of Finance,
Vol. 16, No. 1, pages 8-37, Mar., 1961.
Appendix
Our mechanism and analysis assume that a player’s true valuation maps subsets of the goods to nonnegative numbers (but we could handle negative valuations as well).10
A
Analysis of Our Mechanism
In what follows, all (individual, collective, and vectors of) strategies are relative to mechanism M.
Lemma 1. For all independent players i and all σi ∈ Σ1i : if i 6= ? and αi? 6= ∅ after Stage 1, then in Stage 3
(that is, when M’s coin toss comes up Tails)
1. i answers YES whenever T Vi (αi? ) > πi? , and
2. i answers NO whenever T Vi (αi? ) < πi? .
Proof. We restrict ourselves to just prove, by contradiction, the first implication (the proof of the second
one is totally symmetric). Define the following properties of an execution of M:
P : i 6= ?, αi? 6= ∅, and T Vi (αi? ) > πi? .
P : i = ?, or αi? = ∅, or T Vi (αi? ) ≤ πi? .
Assume that there exists an independent player i and a strategy profile σ ∈ Σ0 such that (1) σi ∈ Σ1i ; (2)
σ’s execution satisfies P; and (3) i answers NO in Stage 3. Consider the following alternative strategy for i:
Strategy σi0
Stage 1. Run σi (with stage input “1” and private inputs T Vi and GKi ) and
announce Ωi and Si as σi does.
Stage 3. If P, run σi and answer whatever σi does.11
If P, answer YES.
We derive a contradiction by proving that σi is dominated by σi0 over Σ0 , which implies that σi 6∈ Σ1i .
Notice that E[ui (M(σi t τ −i ))] = E[ui (M(σi0 t τ −i ))] for all subprofiles τ −i ∈ Σ0−i such that the execution
of σi t τ −i either satisfies (1) P, or (2) P and i answers YES in Stage 3. (This is so because for such τ −i the
executions of σi t τ −i and σi0 t τ −i coincide, and so do their outcomes whenever the coin toss of M is the
same.) Therefore to prove that σi is dominated by σi0 over Σ0 , it suffices to consider the strategy subprofiles
τ−i ∈ Σ0−i such that the execution of σi t τ−i satisfies P and i answers NO in Stage 3. (Notice that, by
assumption, τ−i = σ−i is one such subprofile.)
For all such τ−i , observe that, since σi0 coincides with σi in Stage 1, the outcome profile Ω is the same in
the executions of σi t τ−i and σi0 t τ−i . Accordingly, the star player too is the same in both executions. Since
(by hypothesis) the execution of σi t τ−i satisfies P, so does the execution of σi0 t τ−i .
We now distinguish two cases, each occurring with probability 1/2.
10
In traditional auctions, valuations are bids, and the seller would immediately dismiss bids associating a subset S of the goods
to a negative number (since he has no intention to assign S to a player and also pay him to accept S). The “bidding process”
of our extensive-form mechanism however asks each player i to announce in Step 1 a subset Si of the goods without mentioning
any value for Si . In principle, therefore, i may have a negative valuation for Si . And leaving things as they stand, i may have
(subtle) reasons to announce such an Si .
11
The first implication of Lemma 1 specifies that i 6= ? and T Vi (αi? ) > πi? . However, a strategy must be specified in all cases,
and thus σi0 must be specified also when P.
12
(1) M’s coin toss comes up Heads.
In this case, because only the star player receives goods, we have
ui (M(σi t τ−i )) = ui (M(σi0 t τ−i )) = 0.
(2) M’s coin toss comes up Tails.
In this case, because by hypothesis, (1) T Vi (αi? ) > πi? , (2) player i answers NO in the execution of
σi t τ−i and (3) i answers YES in the execution of σi0 t τ−i , we have
ui (M(σi t τ−i )) = 0
and
ui (M(σi0 t τ−i )) = T Vi (αi? ) − πi? > 0.
Combining the above two cases yields
E[ui (M(σi t τ−i ))] < E[ui (M(σi0 t τ−i ))].
Therefore σi is dominated by σi0 over Σ0 .
Lemma 2. For all minimally monotone collusive sets C and all σC ∈ Σ1C : if ? 6∈ C after Stage 1, then in
Stage 3, for all players i in C
1. i answers YES whenever αi? 6= ∅ and T Vi (αi? ) > πi? , and
2. i answers NO whenever αi? 6= ∅ and T Vi (αi? ) < πi? .
Proof. We again restrict ourselves to just prove the first implication, and proceed by contradiction. Assume
that there exist a minimally monotone collusive set C, a player i ∈ C, and a strategy vector σ such that
σC ∈ Σ1C , σ−C ∈ Σ0−C , and in σ’s execution i answers NO in Stage 3 and the following property holds:
Pi,C : ? 6∈ C, αi? 6= ∅, and T Vi (αi? ) > πi? .
Then, denoting by Pi,C the negation of Pi,C , that is,
Pi,C : ? ∈ C, or αi? = ∅, or T Vi (αi? ) ≤ πi? ,
consider the following alternative collective strategy for C.
0
Strategy σC
Stage 1. Run σC and announce Ωj and Sj as σC does for all j ∈ C.
Stage 3. If Pi,C , continue running σC and answer whatever σC does for all j ∈ C.
If Pi,C , continue running σC , answer YES for i and whatever σC does for all j ∈ C \ {i}.
0 over Σ0 , which implies σ 6∈ Σ1 .
We derive a contradiction by proving that σC is dominated by σC
C
C
0
Similar to Claim 1, to prove that σC is dominated by σC over Σ0 , it suffices to consider all strategy
sub-vectors τ−C ∈ Σ0−C such that the execution of σC t τ−C satisfies Pi,C and i answers NO in Stage 3. (Note
that by hypothesis, τ−C = σ−C is one such strategy sub-vector.) For each such τ−C , we have that (1) for all
0 tτ
0
j ∈ C \{i}, (Ma (σC tτ−C )j , Mp (σC tτ−C )j ) = (Ma (σC
−C )j , Mp (σC tτ−C )j ), and (2) Ma (σC tτ−C )i = ∅
and Mp (σC t τ−C )i = 0, no matter what the coin toss of M is. Thus, due to C’s minimal monotonicity, to
0 tτ
0
show that E[uC (M(σC t τ−C ))] < E[uC (M(σC
−C ))] it suffices to prove that ui (M(σC t τ−C )) = 0 when
0 tτ
the coin toss of M comes up Heads, and that ui (M(σC
−C )) > 0 when the coin toss of M comes up Tails.
This proof is analogous to the corresponding one of Claim 1, and is ignored.
Lemma 3. ∀ independent player i and ∀ σi ∈ Σ2i , rev(Ωi ) ≥ rev(RKi ) (that is, i does not “underbid”).
Proof. We proceed by contradiction. Assume that there exists an independent player i and a strategy
σi ∈ Σ2i such that rev(Ωi ) < rev(RKi ). Now consider the following alternative strategy for player i.
13
Strategy σ
bi
b i = (b
Stage 1. Announce the outcome Ω
αi , π
bi ) = RKi , and
the subset of goods Sbi = arg maxS⊆G T Vi (S).
Stage 3. Announce YES, NO, or the empty string as follows:
If ? = i or αi? = ∅, announce the empty string.
Else, announce YES if T Vi (αi? ) ≥ πi? , and announce NO if T Vi (αi? ) < πi? .
We derive a contradiction in two steps, that is by proving two separate claims: namely, (1) σ
bi ∈ Σ1i , and
(2) σi is dominated by σ
bi over Σ1C for all minimally monotone collusive contexts C compatible with i. The
second fact of course contradicts the assumption that σi ∈ Σ2i .
Claim 1: σ
bi ∈ Σ1i .
bi and σ i dominates σ
bi over Σ0 .
Proof: Proceeding by contradiction, let σ i be a strategy such that σ i 6= σ
i
b i or S i 6= Sbi , and let σ−i be the subprofile of strategies in which every player
Assume that σ i announces Ω 6= Ω
j
bi
j ∈ −i announces Ω such that rev(Ωj ) = 0 and Sj = ∅ in Stage 1, announces YES in Stage 3 if Ω? = Ω
and S ? = Sbi , and NO otherwise. Notice that σ−i clearly belongs to Σ0−i . (Indeed Σ0 consists of what all that
the players can do, independent of any rationality consideration.) Notice too however, since rev(RKi ) > 0
by hypothesis, i = ? under the profile σ
bi t σ−i and E[ui (M(b
σi t σ−i ))] >
maxS⊆G T Vi (S)
E[ui (M(σ i t σ−i ))] ≤
.
2
Accordingly, if σ i dominates σ
bi ,
bi )
T Vi (S
2
=
maxS⊆G T Vi (S)
.
2
While
Therefore such a σ i can not dominate σ
bi over Σ0 .
it must be that σ i announces the same outcome and the same subset
of goods as σ
bi does, and thus coincides with σ
bi in Stage 1. If the coin toss of M comes up Heads, then
the final outcomes under the profiles σ
bi t σ−i and σ i t σ−i are clearly the same, so are ui (M(b
σi t σ−i ))
and ui (M(σ i t σ−i )). Let us now consider the case when the coin toss of M comes up Tails and the two
executions run into Stage 3. There, Lemma 1 implies that the only possible difference between σ
bi and a
dominating σ i consists of what the two strategies announce when i 6= ?, αi? 6= ∅ and T Vi (αi? ) = πi? : namely,
σ
bi answers YES (by definition) and σ i answers NO (because it must be different from σ
bi ). But this syntactic
difference does not translate into any utility difference: indeed, accepting a subset of goods and paying what
your true valuation for it or receiving no goods at all and paying nothing is equivalent. Therefore no σ i 6= σ
bi
can dominate σ
bi over Σ0 . In sum, σ
bi ∈ Σ1i as we wanted to show. Claim 2: For all minimally monotone collusive contexts C compatible with i, σ
bi dominates σi over Σ1C .
Proof: To prove our claim we need to compare E[ui (M(σi t τ−i ))] and E[ui (M(b
σi t τ−i ))] for all strategy
subprofiles τ−i ∈ Σ1C\{i} , where C denotes the player partition of C . Arbitrarily fixing such a τ−i , denoting
b j = (b
αj , π
bj ) the outcomes respectively announced by a player j in the executions of
by Ωj = (αj , π j ) and Ω
b0 respectively the second highest revenue in the two executions,
σi tτ−i and σ
bi tτ−i , and denoting by R0 and R
the following four simple observations hold.
bj.
O1 : ∀j ∈ −i, Ωj = Ω
O2 : If i 6= ? in both executions, then the star player is the same in both executions.
b0 .
O3 : If i = ? in both executions, then R0 = R
O4 : If i = ?, then in Stage 3, each player j offered some goods in the outcome announced by player i answers
YES if his true valuation for these goods is greater than his price in such outcome, and NO if it is less.
(O1 holds because outcomes are announced in Stage 1 where all players act simultaneously without receiving
any information at all from the mechanism M; O2 and O3 are immediate implications of O1 ; and O4 follows
from Lemmas 1 and 2, and the fact that i does not belong to any collusive set.)
To establish that σ
bi dominates σi over Σ1C , we analyze the following four exhaustive cases, again after
arbitrarily fixing τ−i ∈ Σ1C\{i} .
14
Case 1: i 6= ? in the execution of σi t τ−i and i 6= ? in the execution of σ
bi t τ−i .
In this case, by observations O1 and O2 , αi? = α
bi? and πi? = π
bi? . There are four sub-cases.
(a) αi? = ∅. In this sub-case we have E[ui (M(σi t τ−i ))] = E[ui (M(b
σi t τ−i ))] = 0.
(b) αi? 6= ∅ and T Vi (αi? ) = πi? . In this sub-case, no matter whether player i answers YES or NO in
Stage 3 of σi , we have E[ui (M(σi t τ−i ))] = E[ui (M(b
σi t τ−i ))] = 0.
(c) αi? 6= ∅ and T Vi (αi? ) < πi? . In this sub-case, by Lemma 1, i answers NO in Stage 3 of both
executions, and we have E[ui (M(σi t τ−i ))] = E[ui (M(b
σi t τ−i ))] = 0.
(d) αi? 6= ∅ and T Vi (αi? ) > πi? . In this sub-case, by Lemma 1, i answers YES in Stage 3 of both
T V (b
α? )−b
πi?
T V (α? )−πi?
= i 2i
= E[ui (M(b
σi t τ−i ))].
executions, and we have E[ui (M(σi t τ−i ))] = i 2i
In sum, no matter which sub-case applies, Case 1 implies E[ui (M(σi t τ−i ))] = E[ui (M(b
σi t τ−i ))].
Case 2: i 6= ? in the execution of σi t τ−i and i = ? in the execution of σ
bi t τ−i .
In this case, let us first prove that E[ui (M(σi t τ−i ))] ≤ T Vi2(Si ) . To this end, we consider the same four
sub-cases as above. Namely,
(a) αi? = ∅. In this sub-case, E[ui (M(σi t τ−i ))] = 0. Therefore, since T Vi (Sbi ) ≥ 0 by definition, we
b
have E[ui (M(σi t τ−i ))] ≤
(b)
αi?
6= ∅ and
T Vi (αi? )
=
πi? .
bi )
T Vi (S
2
as desired.
In this sub-case, no matter whether player i answers YES or NO in
Stage 3, we also have E[ui (M(σi t τ−i ))] = 0, and thus E[ui (M(σi t τ−i ))] ≤
bi )
T Vi (S
.
2
(c) αi? 6= ∅ and T Vi (αi? ) < πi? . In this sub-case, player i answers NO in Stage 3, and thus E[ui (M(σi t
τ−i ))] = 0 ≤
bi )
T Vi (S
.
2
?
T Vi (αi )
(d) αi? 6= ∅ and
> πi? . In this sub-case, player i answers YES in Stage 3, causing himself to
be assigned the subset of goods αi? for price πi? , and thus can have positive utility. Accordingly
T V (α?i )−πi?
2
i
E[ui (M(σi t τ−i ))] =
T Vi (Sbi ) = maxS⊆G T Vi (S).
≤
T Vi (α?i )
2
≤
bi )
T Vi (S
.
2
In fact, πi? is always non-negative, and
In sum, no matter which sub-case applies, we have E[ui (M(σi t τ−i ))] ≤
bi )
T Vi (S
.
2
σi t τ−i ))]. In this case, i’s expected utility in the execution
Let us now prove that T Vi2(Si ) ≤ E[ui (M(b
of σ
bi t τ−i is the weighted sum of his utilityPwhen M’s coin toss isPHeads and his utility when M’s
”) the sum taken over every
” (respectively,“ j:N
coin toss is Tails.12 Therefore, denoting by “ j:Y[
d
O
ES
player j who answers YES (respectively, NO) in Stage 3 of the execution of σ
bi t τ−i (that is, M’s coin
toss comes up Tails), we have
P
P
b0
π
bi − j:N
π
bi − R
d
T Vi (Sbi )
O j
j:Y[
ES j
E[ui (M(b
σi t τ−i ))] =
+
.
2
2
b
By definition of RKi and compatibility, ∀j ∈ −i such that α
bji 6= ∅, π
bi < T Vj (b
αji ). Thus by observation
P j
P i
P
O4 every such player j answers YES in Stage 3: in our notation j:Y[
π
bi = 0.
π
bi = j π
bj and j:N
d
O j
ES j
Accordingly, we have
P i
b0
b i) − R
b0
bj − R
T Vi (Sbi ) + rev(Ω
T Vi (Sbi )
jπ
+
=
.
E[ui (M(b
σi t τ−i ))] =
2
2
2
b i) ≥ R
b0 , we have T Vi (Si ) ≤ E[ui (M(b
Since rev(Ω
σi t τ−i ))] as desired.
2
Therefore Case 2 implies E[ui (M(σi t τ−i ))] ≤ E[ui (M(b
σi t τ−i ))].
b
Case 3: i = ? in the execution of σi t τ−i and i = ? in the execution of σ
bi t τ−i .
In this case, similar to Case 2, i’s expected utility in the execution of σi t τ−i is the weighted sum of
his utility when M’s coin toss is Heads and his utility when M’s coin toss is Tails. Therefore, denoting
12
Both individual utilities are expected, if the strategies of the other players are probabilistic.
15
P
P
by “ j:Y ES ” (respectively,“ j:N O ”) the sum taken over every player j who answers YES (respectively,
NO) in Stage 3 of the execution of σi t τ−i , we have
P
P
i
i
0
T Vi (Si )
j:Y ES πj −
j:N O πj − R
E[ui (M(σi t τ−i ))] =
+
.
2
2
P
Since j:N O πji ≥ 0, we have that
P i
0
T Vi (Si ) + rev(Ωi ) − R0
T Vi (Si )
j πj − R
+
=
.
E[ui (M(σi t τ−i ))] ≤
2
2
2
Let us now analyze i’s expected utility in the execution of σ
bi tτ−i . Same as in Case 2, and by observation
O3 , we have that
E[ui (M(b
σi t τ−i ))] =
b i) − R
b0
b i ) − R0
T Vi (Sbi ) + rev(Ω
T Vi (Sbi ) + rev(Ω
=
.
2
2
b i ) = rev(RKi ) > rev(Ωi ); and (2)
According to our construction of σ
bi , we have that: (1) rev(Ω
T Vi (Sbi ) = maxS⊆G T Vi (S). Therefore
E[ui (M(b
σi t τ−i ))] >
T Vi (Si ) + rev(Ωi ) − R0
.
2
In sum, Case 3 implies E[ui (M(σi t τ−i ))] < E[ui (M(b
σi t τ−i ))].
Case 4: i = ? in the execution of σi t τ−i and i 6= ? in the execution of σ
bi t τ−i .
i
i
b
Fortunately, this case can never happen. Since rev(Ω ) > rev(Ω ) (by construction) and ∀j ∈ −i Ωj =
b j (by observation O1 ), we have that if i = ? in the execution of σi t τ−i , it must be true that i = ?
Ω
also in the execution of σ
bi t τ−i .
Having finished to analyze all possible cases, we conclude that σi is dominated by σ
bi over Σ1C .
Since both Claims 1 and 2 hold, so does Lemma 3.
Comment. Note that, while ruling out the underbidding (relative to RKi ) of independent players, our
analysis says nothing about the possibility of “over-bidding.” In fact, assume that player i’s general knowledge
GKi includes some Bayesian information about the true valuation of another player j that enables i to
compute the probability that T Vj (S) > v for some subset S of goods and a particular value v. Then,
depending on such probability and v, rather then announcing the outcome Ωi = RKi , i may be better off
announcing Ωi such that αji = S, πji > v, and rev(Ωi ) > rev(RKi ) (taking into account the probability that
j may reject this offer.) Therefore over-bidding may not be a dominated strategy for player i over Σ1C . But
as shown in the following proof, if a player over-bids, our result still holds, and thus we do not care whether
over-bidding is dominated or not.
Finally, let us now (recall and) prove our main theorem.
Theorem 1. For all minimally monotone collusive contexts C and all Σ1 /Σ2I plays σ of (C , M),
E[rev(M(σ))] + E[sw(M(σ), T V )] ≥
MEW(RKI )
,
2
(and E[rev(M(σ))] ≥
R0
).
2
Proof. Denote by ∗ the independent player “realizing” our benchmark: that is,
∗ = arg max rev(RKi ).
i∈I
Notice that the players ∗ and ? need not to coincide, and notice that the following two inequalities hold in
any Σ1 /Σ2I play of (C , M):
16
(a) rev(Ω∗ ) ≥ rev(RK∗ ) = MEW(RKI ).
(b) R? ≥ MEW(RKI ).
Indeed, Inequality (a) holds because player ∗ is independent and, by Lemma 3, he does not underbid; and
Inequality (b) holds by Inequality (a) and the fact that R? ≥ rev(Ω∗ ) by the very definition of the star
player.
To prove our theorem, we distinguish two cases.
Case 1: ? = ∗.
In this case, as player ∗ is independent, so is player ?, and thus ? 6∈ C for all collusive sets C in the
player partition of C . Therefore Lemma 1 and 2 guarantee that when M’s coin toss comes up Tails,
every i 6= ? answers YES only if T Vi (αi? ) ≥ πi? . Accordingly, the following inequality holds for M’s
expected social welfare:
P
P
P
?
?
π?
T V? (S? )
i:Y ES T Vi (αi )
i:Y ES T Vi (αi )
E[sw(M(σ), T V )] =
+
≥
≥ i:Y ES i .
(1)
2
2
2
2
At the same time,
P
R0 + i:N O πi?
E[rev(M(σ))] =
,
2
which immediately translates into the following two inequalities:
E[rev(M(σ))] ≥
and
P
E[rev(M(σ))] ≥
R0
2
i:N O
(2)
πi?
.
(3)
2
Thus, in the case at hand, Inequality (2) coincides with the second part of our thesis, while the first
part follows by combining Inequalities (1) and (3) and then using Inequality (b): namely,
P
P
P ?
?
?
π
R?
MEW(RKI )
i:Y ES πi +
i:N O πi
E[sw(M(σ), T V )] + E[rev(M(σ))] ≥
= i i =
≥
.
2
2
2
2
Case 2: ? 6= ∗.
In this case, we have ∗ ∈ −?, and thus R0 ≥ rev(Ω∗ ) by the definition of R0 . Therefore, by Inequality
(a), we have
P
R0 + i:N O πi?
R0
MEW(RKI )
E[rev(M(σ))] =
≥
≥
.
2
2
2
Of course
P
T Vi (αi? )
T V? (S? )
+ i:Y ES
≥ 0.
E[sw(M(σ), T V )] =
2
2
Thus summing term by term we have
E[sw(M(σ), T V )] + E[rev(M(σ))] ≥
Q.E.D.
17
MEW(RKI )
.
2